The Existence/Uniqueness of Solutions to Higher Order Linear Differential Equations

We will now begin to look at methods to solving higher order differential equations.

Definition: An $n^{\mathrm{th}}$ Order Differential Equation is a differential equation containing an $n^{\mathrm{th}}$ derivative and can be written in the form $\frac{d^n y}{dt^n} = f \left (t, y, \frac{dy}{dt}, ..., \frac{d^{(n-1)}y}{dt^{(n-1)}} \right )$.

Expanded out, we have that an $n^{\mathrm{th}}$ order linear differential equation has the form:

Once again, it is important to stress that Theorem 1 above is simply an extension to the Theorems on the existence and uniqueness of solutions to first order and second order linear differential equations. Furthermore, for this theorem to apply, we must have that coefficient in front of the $\frac{d^n}{dt^n}$ term is $1$.

Let's look at an example of verifying that a unique solution to a higher order linear differential equation exists.

Note that the functions $p_1(t) = \frac{3}{t}$, $p_2(t) = \frac{\sin t}{t}$, and $p_3(t) = \frac{e^t}{t}$ are all continuous for $t \neq 0$. However, we have that $t_0 = 1 \neq 0$, and so in fact we have that a unique solution $y = \phi(t)$ exists to this third order linear differential equation through the interval $(0, \infty)$.